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Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

By Jung JongSoo

Abstract

<p>Abstract</p> <p>Let <inline-formula> <graphic file="1687-1812-2008-167535-i1.gif"/></inline-formula> be a reflexive Banach space with a uniformly G&#226;teaux differentiable norm. Suppose that every weakly compact convex subset of <inline-formula> <graphic file="1687-1812-2008-167535-i2.gif"/></inline-formula> has the fixed point property for nonexpansive mappings. Let <inline-formula> <graphic file="1687-1812-2008-167535-i3.gif"/></inline-formula> be a nonempty closed convex subset of <inline-formula> <graphic file="1687-1812-2008-167535-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i5.gif"/></inline-formula> a contractive mapping (or a weakly contractive mapping), and <inline-formula> <graphic file="1687-1812-2008-167535-i6.gif"/></inline-formula> nonexpansive mapping with the fixed point set <inline-formula> <graphic file="1687-1812-2008-167535-i7.gif"/></inline-formula>. Let <inline-formula> <graphic file="1687-1812-2008-167535-i8.gif"/></inline-formula> be generated by a new composite iterative scheme: <inline-formula> <graphic file="1687-1812-2008-167535-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i10.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i11.gif"/></inline-formula>. It is proved that <inline-formula> <graphic file="1687-1812-2008-167535-i12.gif"/></inline-formula> converges strongly to a point in <inline-formula> <graphic file="1687-1812-2008-167535-i13.gif"/></inline-formula>, which is a solution of certain variational inequality provided that the sequence <inline-formula> <graphic file="1687-1812-2008-167535-i14.gif"/></inline-formula> satisfies <inline-formula> <graphic file="1687-1812-2008-167535-i15.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1812-2008-167535-i16.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i17.gif"/></inline-formula> for some <inline-formula> <graphic file="1687-1812-2008-167535-i18.gif"/></inline-formula> and the sequence <inline-formula> <graphic file="1687-1812-2008-167535-i19.gif"/></inline-formula> is asymptotically regular.</p

Topics: Mathematics, QA1-939, Science, Q, DOAJ:Mathematics, DOAJ:Mathematics and Statistics, Applied mathematics. Quantitative methods, T57-57.97, Analysis, QA299.6-433
Publisher: Springer
Year: 2008
OAI identifier: oai:doaj.org/article:2a6368be742d47e7b4d6b9a794f3b013
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